The conjectured $B_n$/$C_n$ symmetry algebra of the periodic Fredkin chain: proof or refutation of Pronko's Conjectures 1–4 and of $[\Xi,\mathrm{H}]=0$

The periodic Fredkin chain — a frustration-free spin-1/2 model with three-site interactions whose degenerate ground states are labeled by Dyck-path equivalence classes — was shown by Pronko (2025) to possess two nonlocal ladder operators $\Sigma^\pm$ that commute with the Hamiltonian and connect its ground states. Exact computations for chains of up to $N=10$ sites indicate that these operators generate a Lie algebra of type $B_n$ (odd $N$) or $C_n$ (even $N$) of rank $n=\lceil N/2\rceil$, extended by a central element contained in the total-spin component $\mathrm{S}^z$; all of these structural claims are conjectural. The problem is to establish the conjectured symmetry algebra rigorously for arbitrary $N$: prove or refute that $\Sigma^\pm$ annihilate every non-cyclic-invariant eigenstate, that $\Sigma^\pm$ are representable through nested anticommutators of the total-spin operators with coefficients $\gamma_k(N)$, that the generated algebra is exactly $B_n$ or $C_n$ in the Chevalley–Serre sense, that $\mathrm{S}^z$ decomposes into Cartan elements plus a central element with coefficients $\alpha_i(N)$, and that the diagonal $S^z=0$-sector operator $\Xi$ commutes with the Hamiltonian for even $N$. A 2025 proof that the chain has no local conserved charges settled the adjacent integrability question negatively, leaving the symmetry-algebra conjectures untouched and open.

Background

The Fredkin chain is a frustration-free spin-1/2 model (a model whose ground states simultaneously minimize every interaction term of the Hamiltonian) introduced by Salberger and Korepin (2017) as a spin-1/2 analogue of the spin-1 Motzkin chain (a sibling frustration-free model of nearest-neighbor-interacting spin-1 spins whose ground state is a uniform sum of Motzkin paths, lattice walks with steps of height $0$ or $\pm 1$ that never cross below the axis). Its Hamiltonian on $N$ sites with periodic boundary conditions is a cyclic sum of three-site densities,

$$H=\sum_{j=1}^{N}\left(n^{\uparrow}_{j}\,\Pi_{j+1,j+2}+\Pi_{j,j+1}\,n^{\downarrow}_{j+2}\right),$$

where $n^{\uparrow}_j$ and $n^{\downarrow}_j$ project site $j$ onto spin up/down and $\Pi_{j,j+1}=\tfrac{1}{2}(1-P_{j,j+1})$ projects the two spins at sites $j,j+1$ onto their spin-0 state ($P$ is the permutation operator swapping the two sites); the interaction descends from the Fredkin (controlled-SWAP) gate of reversible computation. With open boundaries the ground state is the uniform superposition of all Dyck paths — lattice walks of $N$ steps of height $\pm 1$ that never cross below the axis — and its entanglement entropy grows logarithmically with subsystem size, violating the area law (the expectation that the entanglement entropy of a local-Hamiltonian ground state scale with the boundary of a subsystem rather than with its volume). With periodic boundary conditions, the case of interest here, the ground-state space is degenerate: a theorem of Salberger and Korepin states that for each eigenvalue $S^z\in\{-N/2,\dots,N/2\}$ of the third component of total spin there is exactly one zero-energy ground state with cyclic-shift (translation) eigenvalue $C=1$, and for even $N$ there is one additional $C=-1$ ground state at $S^z=0$. Each ground state is a sum over an equivalence class of Dyck-path configurations, two paths being equivalent when they differ by moving “peaks” (Fredkin moves) along the path. In the notation of the source, $C_{a,b}(N)$ (with $N=a+2k+b$, $k\ge 0$) denotes the class of length-$N$ paths with steps of height $\pm 1$ obtained by Fredkin moves from the reference path consisting of $a$ down steps, $k$ peaks and $b$ up steps; its basis vectors are $S^z$-eigenvectors with eigenvalue $\tfrac{1}{2}(b-a)$, and for even $N$ the class $C_{0,0}(N)$ is the standard Dyck class of walks that never cross below the axis.

Two exact symmetries are obvious: the cyclic shift operator $\mathrm{C}$ and $\mathrm{S}^z$ commute with $H$. Pronko (2025) discovered a much larger hidden symmetry. He defined the two nonlocal operators

$$\Sigma^{\pm}=\sum_{\substack{r_1,\dots,r_N\in\{-1,0,1\}\\ r_1+\dots+r_N=\pm 1}}\sigma_1^{r_1}\cdots\sigma_N^{r_N},$$

with $\sigma_i^0=1$ and $\sigma_i^{\pm 1}=\sigma_i^{\pm}$ the Pauli raising/lowering matrices (equivalently, the residue at $\lambda=0$ of $\prod_{i=1}^{N}(\lambda^{-1}\sigma_i^{\pm}+1+\lambda\sigma_i^{\mp})$), and proved rigorously (his Theorem 2) that $[\Sigma^{\pm},H]=0$ while $\Sigma^{\pm}$ act as ladder operators on the ground states, mapping the ground state at a given $S^z$ to that at $S^z\pm 1$. All of his further structural findings are conjectural, based on exact computations for $N\le 10$:

• Conjecture 1: $\Sigma^{\pm}$ annihilate every eigenstate of $H$ that is not cyclic-invariant (i.e., with $C\ne 1$), so the hidden symmetry acts nontrivially only on the $C=1$ subspace.
• Conjecture 2: $\Sigma^{\pm}$ admit a representation entirely in terms of the total-spin operators $\mathrm{S}^{\pm}=\sum_j\sigma_j^{\pm}$ via nested anticommutators,

$$\Sigma^{\pm}=\sum_{k=1}^{\lceil N/2\rceil}\gamma_k\left(\mathrm{aad}\,\mathrm{S}^{\pm}\,\mathrm{aad}\,\mathrm{S}^{\mp}\right)^{k-1}\mathrm{S}^{\pm},$$

where $\mathrm{aad}$ is the anti-adjoint action, $(\mathrm{aad}\,a)b\equiv\{a,b\}=ab+ba$, and the rational coefficients $\gamma_k$ depend on $N$ (for example, for $N=3$: $\Sigma^{\pm}=-\tfrac{1}{4}\mathrm{S}^{\pm}+\tfrac{1}{8}\{\mathrm{S}^{\pm},\{\mathrm{S}^{\mp},\mathrm{S}^{\pm}\}\}$).

• Conjecture 3: the Lie algebra $\mathfrak{g}$ generated by $\Sigma^{\pm}$ has rank $n=\lceil N/2\rceil$ and is of type $B_n=\mathfrak{so}_{2n+1}$ for odd $N=2n-1$ or $C_n=\mathfrak{sp}_{2n}$ for even $N=2n$ — the odd-dimensional orthogonal and the symplectic Lie algebras — meaning that Chevalley generators $e_i,f_i,h_i$ (the standard generators of a semisimple Lie algebra, defined by the commutation and Serre relations with a given Cartan matrix, the encoding of the algebra's Dynkin diagram) can be built from $\Sigma^{\pm}$ and its iterated commutators and satisfy the Serre relations with the $B_n$/$C_n$ Cartan matrix.
• Conjecture 4: the symmetry algebra is extended by a central element $p$ contained in the total-spin component: $\mathrm{S}^{z}=p+\sum_{i=1}^{\lceil N/2\rceil}\alpha_i h_i$, with $h_i$ the Cartan subalgebra elements (a basis of the maximal commuting subalgebra of $\mathfrak{g}$) and coefficients $\alpha_i$ depending on $N$; the empirically observed patterns include $\alpha_1=N/2$ for $N\ge 4$ and $\alpha_2=N-1$ for $N\ge 5$.
• For even $N$, a further operator $\Xi$ (equations (4.1)–(4.3) of that paper) — diagonal, acting only in the $S^z=0$ sector with $0,+1,-1$ entries organized by the Dyck-path classes $C_{k,k}(N)$ — exchanges the two $S^z=0$ ground states, anti-commutes with the cyclic shift (obvious from its form), and appears to commute with $H$ in all checks; the proof of $[\Xi,H]=0$ is lacking, and $\Xi$ would explain a clearly visible double degeneracy of the whole spectrum.

These conjectures are the spin-1/2 analogues of those made earlier by the same author for the periodic spin-1 Motzkin chain, where the algebra is conjectured to be of $C$-type with rank equal to the number of sites; remarkably, the central-element formula for even $N$ coincides exactly with the Motzkin-chain formula, suggesting a structure independent of the spin representation. The picture is known not to be an integrability statement in the usual sense: Fan et al. (2025) proved that the Fredkin chain, periodic or open, admits no local conserved charges (operators supported on a fixed finite number of adjacent sites that commute with $H$) beyond the trivial ones, so the conjectured $B_n/C_n$ symmetry is a genuinely nonlocal symmetry of a nonintegrable chain (one lacking the infinite family of conserved quantities characteristic of Yang–Baxter-integrable models). What is missing is any rigorous general-$N$ treatment of the conjectures themselves: everything rests on exact computations up to $N=10$, and the intermediate equations needed to identify the Cartan matrix (solving quadratic systems for the Chevalley generators) already involve integers of roughly two hundred digits at $N=10$, blocking direct numerical extension.

Problem Statement

Determine, with rigorous proof valid for arbitrary chain length $N$, whether the conjectured hidden symmetry structure of the periodic Fredkin chain holds; specifically, prove or refute each of the following claims:

1. (Conjecture 1 of Pronko 2025) The operators $\Sigma^{\pm}$ annihilate all eigenstates of the periodic Fredkin Hamiltonian $H$ with cyclic-shift eigenvalue $C\ne 1$.
2. (Conjecture 2) The operators $\Sigma^{\pm}$ admit the representation $\Sigma^{\pm}=\sum_{k=1}^{\lceil N/2\rceil}\gamma_k(\mathrm{aad}\,\mathrm{S}^{\pm}\,\mathrm{aad}\,\mathrm{S}^{\mp})^{k-1}\mathrm{S}^{\pm}$ with coefficients $\gamma_k$ depending on $N$.
3. (Conjecture 3) The Lie algebra $\mathfrak{g}$ generated by $\Sigma^{\pm}$ has rank $n=\lceil N/2\rceil$ and is exactly of type $B_n$ for odd $N=2n-1$ and $C_n$ for even $N=2n$: Chevalley generators $e_i,f_i,h_i$ can be built from $\Sigma^{\pm}$ and its iterated commutators and satisfy the Serre relations with the $B_n$/$C_n$ Cartan matrix.
4. (Conjecture 4) There exists a nontrivial central element $p=\mathrm{S}^{z}-\sum_{i=1}^{\lceil N/2\rceil}\alpha_i h_i$ with respect to the symmetry algebra $\mathfrak{g}$ (i.e. $[p,\mathfrak{g}]=0$), with coefficients $\alpha_i$ depending on $N$.
5. ($\Xi$-commutativity) For even $N$, the diagonal operator

$$\Xi=\sum_{k=0}^{N/2}(-1)^{k}\sum_{\ell_1\ell_2\dots\ell_N\in C_{k,k}(N)}n_1^{\ell_1}n_2^{\ell_2}\cdots n_N^{\ell_N},$$

with $n_j^{\ell_j}$, $\ell_j=\uparrow,\downarrow$, the projectors of site $j$ onto spin up/down — so that the entries of $\Xi$ are $(-1)^k$ on the Dyck-path class $C_{k,k}(N)$ and $0$ elsewhere, $\Xi$ acting only in the $S^z=0$ sector (equation (4.3) of Pronko 2025) — satisfies $[\Xi,H]=0$.

The common objective uniting these claims is to replace the exact numerical evidence for $N\le 10$ with general theorems establishing the exact symmetry algebra of the periodic Fredkin chain and its action on the spectrum. Any method is admissible — direct operator algebra, representation-theoretic construction, or identification of the $\Sigma^{\pm}$-generated algebra inside a known algebraic structure acting on Dyck-path space — and, since each claim is a universal statement over $N$, a single verified counterexample at any specific $N$ refutes the corresponding claim. A full structural answer is expected to illuminate the group-theoretical origin of the $B$/$C$ dichotomy and the combinatorial (Dyck-path) meaning of the coefficient sequences; the completion boundary of the problem is the determination — proof or refutation — of claims (1)–(5).

References

1. Andrei G. Pronko, Symmetries of the periodic Fredkin chain, Journal of Physics A: Mathematical and Theoretical 58 (2025) 445204, doi:10.1088/1751-8121/ae1644, arXiv:2507.18291, https://arxiv.org/abs/2507.18291
2. Andrei G. Pronko, Periodic Motzkin chain: Ground states and symmetries, Nuclear Physics B 1017 (2025) 116963, doi:10.1016/j.nuclphysb.2025.116963, arXiv:2504.00835, https://arxiv.org/abs/2504.00835
3. Wen-Ming Fan, Kun Hao, Yang-Yang Chen, Kun Zhang, Xiao-Hui Wang, Vladimir Korepin, Absence of local conserved charges of the Fredkin spin chain and its truncated versions, Physical Review B 112 (2025) 205124, doi:10.1103/2qhs-fz3g, arXiv:2509.04838, https://arxiv.org/abs/2509.04838
4. Shunsuke Sengoku, Haruki Watanabe, Absence of nontrivial local conserved quantities in a class of $U(1)$-symmetric spin-1 chains, arXiv:2608.17548 (2026), https://arxiv.org/abs/2608.17548
5. Olof Salberger, Vladimir Korepin, Entangled spin chain (arXiv title: Fredkin Spin Chain), Reviews in Mathematical Physics 29 (2017) 1750031, doi:10.1142/S0129055X17500313, arXiv:1605.03842, https://arxiv.org/abs/1605.03842